Invariant Sets of Hamiltonian Systems and Variational Methods

Sergey V. Bolotin
Additional contact information
Sergey V. Bolotin: Moscow State University, Department of Mathematics and Mechanics

A chapter in Proceedings of the International Congress of Mathematicians, 1995, pp 1169-1178 from Springer

Abstract: Abstract We study the problem on the existence of homoclinic trajectories to Mather minimizing invariant sets (multidimensional generalization of Aubry-Mather sets) of positive definite time-periodic Hamiltonian systems [19]. These sets are supports of invariant probability measures in the Lagrangian L. For natural systems with L = ǁυǁ2/2 — V(x), the minimizing set is Γ = {V = h}, h = max V, and for time-periodic systems with reversible L the minimizing sets consist of brake orbits of minimal action. For natural Hamiltonian systems, the existence of homoclinics to Γ was proved in [3] using the Maupertuis-Jacobi functional $$ \smallint \sqrt {h - V(x)} \parallel dx\parallel ,$$ and for reversible time-periodic systems in [4] using Hamilton’s functional (see also [5], [16]). For nonreversible systems (for example, natural systems with gyroscopic forces), in general there are no Mather sets of simple structure. We extend the above existence results to arbitrary minimizing sets replacing homoclinic trajectories by semihomoclinic ones in Birkhoff’s sense [2]. A similar problem was studied in [20].

Date: 1995
References: Add references at CitEc
Citations:

There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-0348-9078-6_110

Ordering information: This item can be ordered from
http://www.springer.com/9783034890786

DOI: 10.1007/978-3-0348-9078-6_110

Access Statistics for this chapter

More chapters in Springer Books from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().